<<

. 51
( 193 .)



>>

as the explanatory variable. Calculate beta using daily beta over a period of one
month and repeat for several months. How stable is the beta estimate?
You can also calculate betas using monthly data from the Monthly Valuation Data
2.
report in Market Insight. Calculate beta using 2 years of monthly data, and repeat
for different periods. Are the 2-year betas more stable than the one month betas
computed in question 1? What are the trade-offs that an analyst might consider
in using beta computed from long-term versus short-term data sets?




WEBMA STER
Returns, Risk, and Correlations
Go to www.mhhe.com/edumarketinsight. Pull the monthly returns for General Electric,
The Home Depot, Johnson & Johnson, Honeywell, and H.J. Heinz. Merge the returns
for these five firms into a single Excel workbook, with the returns for each company
properly aligned.
Then,
1. Using the Excel functions for average and standard deviation (sample), calculate
the average and standard deviation for each of the firms.
2. Using the correlation function, construct the correlation matrix for the five funds
www.mhhe.com/bkm



based on their monthly returns for the entire period. What are the lowest and
highest individual pair of correlation coefficients?
Bodie’Kane’Marcus: II. Portfolio Theory 6. Efficient Diversification © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition




206 Part TWO Portfolio Theory


SOLUTIONS TO 1. Recalculation of Spreadsheets 6.1, 6.2, and 6.4 shows that the correlation coefficient with the new
rates of return is .98.

>
Concept A B C D E F
Stock Fund Bond Fund
CHECKS 1
Scenario Probability Rate of Return Col. B Col. C Rate of Return Col. B Col. E
2
Recession 0.3 -12 -3.6 10 3
3
Normal 0.4 10 4 7 2.8
4
Boom 0.3 28 8.4 2 0.6
5
Expected or Mean Return SUM: 8.8 SUM: 6.4
6
7
8
Stock Fund Bond Fund
9
Squared Deviations Squared Deviations
10
Scenario Probability from Mean Col. B Col. C from Mean Col. B Col. E
11
Recession 0.3 432.64 129.792 12.96 3.888
12
Normal 0.4 1.44 0.576 0.36 0.144
13
Boom 0.3 368.64 110.592 19.36 5.808
14
SUM: SUM:
Variance = 240.96 9.84
15
Std Dev = Variance 15.52 3.14
16
17
Deviation from Mean Return Covariance
18
Scenario Probability Stock Fund Bond Fund Product of Dev Col. B Col. E
19
Recession 0.3 -20.8 3.6 -74.88 -22.464
20
Normal 0.4 1.2 0.6 0.72 0.288
21
Boom 0.3 19.2 -4.4 -84.48 -25.344
22
Covariance: SUM: -47.52
23
Correlation coefficient = Covariance/(StdDev(stocks)*StdDev(bonds)): -0.98
24




2. a. Using Equation 6.3 with the data: 12; 25; wB 0.5; and wS 1 wB 0.5, we
B S
obtain the equation
2
152 2 2
(wB B) (wS S) 2(wB B)(wS S) BS
P

12)2 25)2
(0.5 (0.5 2(0.5 12)(0.5 25) BS

which yields 0.2183.
b. Using Equation 6.2 and the additional data: E(rB) 6; E(rS) 10, we obtain
E(rP) wBE(rB) wSE(rS) (0.5 6) (0.5 10) 8%
c. On the one hand, you should be happier with a correlation of 0.2183 than with 0.22 since the
lower correlation implies greater benefits from diversification and means that, for any level of
expected return, there will be lower risk. On the other hand, the constraint that you must hold
50% of the portfolio in bonds represents a cost to you since it prevents you from choosing the
risk-return trade-off most suited to your tastes. Unless you would choose to hold about 50% of
the portfolio in bonds anyway, you are better off with the slightly higher correlation but with the
www.mhhe.com/bkm




ability to choose your own portfolio weights.
3. The scatter diagrams for pairs B“E are shown below. Scatter diagram A shows an exact conflict
between the pattern of points 1,2,3 versus 3,4,5. Therefore the correlation coefficient is zero.
Scatter diagram B shows perfect positive correlation (1.0). Similarly, C shows perfect negative
correlation ( 1.0). Now compare the scatters of D and E. Both show a general positive correlation,
but scatter D is tighter. Therefore D is associated with a correlation of about .5 (use a spreadsheet to
show that the exact correlation is .54) and E is associated with a correlation of about .2 (show that
the exact correlation coefficient is .23).
Bodie’Kane’Marcus: II. Portfolio Theory 6. Efficient Diversification © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition




207
6 Efficient Diversification


Scatter diagram C
Scatter diagram B

6
6
5
5
4
4




Stock 2
Stock 2




3
3
2
2
1
1
0
0
0 1 2 3 4 5 6
0 1 2 3 4 5 6
Stock 1
Stock 1


Scatter diagram E
Scatter diagram D

6
6
5
5
4
4
Stock 2
Stock 2




3
3
2
2
1
1
0
0
0 1 2 3 4 5 6
0 1 2 3 4 5 6
Stock 1
Stock 1




www.mhhe.com/bkm
Bodie’Kane’Marcus: II. Portfolio Theory 6. Efficient Diversification © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition




208 Part TWO Portfolio Theory




4. a. Implementing Equations 6.2 and 6.3 we generate data for the graph. See the spreadsheet above.
b. Implementing the formulas indicated in the following spreadsheet we generate the optimal risky
portfolio (O) and the minimum variance portfolio.
c. The slope of the CAL is equal to the risk premium of the optimal risky portfolio divided by its
www.mhhe.com/bkm




standard deviation, (11.28 5)/17.59 .357.
d. The mean of the complete portfolio .2222 11.28 .7778 5 6.40% and its standard
deviation is .2222 17.58 3.91%.
The composition of the complete portfolio is
.2222 .26 .06 (i.e., 6%) in X
.2222 .74 .16 (i.e., 16%) in M
and 78% in T-bills.
5. Efficient frontiers derived by portfolio managers depend on forecasts of the rates of return on
various securities and estimates of risk, that is, standard deviations and correlation coefficients.
The forecasts themselves do not control outcomes. Thus, to prefer a manager with a rosier forecast
(northwesterly frontier) is tantamount to rewarding the bearers of good news and punishing the
Bodie’Kane’Marcus: II. Portfolio Theory 6. Efficient Diversification © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition




209
6 Efficient Diversification


Optimal risky portfolio
25



CAL
20
Portfolio mean (%)




15 Efficient frontier of
risky assets
O X

Min. Var. Pf
10

M


5 C


<<

. 51
( 193 .)



>>